Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127265 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(18,23,24,27)(19,20)(21,29)(25,32,31,26)(28,30);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,18,20,25,29,32,19,27,22,24,30,31,21,26,28,23);(18,24)(23,27)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=ghi,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ci,d^b=di,e^b=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240742, #72240743, #72240745, … |
| maximal normal subgroup of | #72240742, #72240743, #72240745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34465);